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Q‐Curves and Abelian Varieties of GL 2 ‐Type
Author(s) -
Quer Jordi
Publication year - 2000
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012570
Subject(s) - mathematics , abelian group , section (typography) , arithmetic of abelian varieties , isogeny , abelian variety , abelian variety of cm type , type (biology) , modular curve , complex multiplication , pure mathematics , elliptic curve , functor , endomorphism , quotient , algebra over a field , elementary abelian group , rank of an abelian group , ecology , biology , advertising , business
The relation between Q‐curves and certain abelian varieties of GL 2 ‐type was established by Ribet (‘Abelian varieties over Q and modular forms’, Proceedings of the KAIST Mathematics Workshop (1992) 53–79) and generalized to building blocks, the higher‐dimensional analogues of Q‐curves, by Pyle in her PhD Thesis (University of California at Berkeley, 1995). In this paper we investigate some aspects of Q‐curves with no complex multiplication and the corresponding abelian varieties of GL 2 ‐type, for which we mainly use the ideas and techniques introduced by Ribet (op. cit. and ‘Fields of definition of abelian varieties with real multiplication’, Contemp.\ Math. 174 (1994) 107–118). After the Introduction, in Sections 2 and 3 we obtain a characterization of the fields where a Q‐curve and all the isogenies between its Galois conjugates can be defined up to isogeny, and we apply it to certain fields of type (2,…,2). In Section 4 we determine the endomorphism algebras of all the abelian varieties of GL 2 ‐type having as a quotient a given Q‐curve in easily computable terms. Section 5 is devoted to a particular case of Weil's restriction of scalars functor applied to a Q‐curve, in which the resulting abelian variety factors over Q up to isogeny as a product of abelian varieties of GL 2 ‐type. Finally, Section 6 contains examples: we parametrize the Q‐curves coming from rational points of the modular curves X *N having genus zero, and we apply the results of Sections 2–5 to some of the curves obtained. We also give results concerning the existence of quadratic Q‐curves. 1991 Mathematics Subject Classification : primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.